This answer is written with the assumption that the querent wishes to fight and generally adventure in the ooze body described in the Fluidic Body entry. That same entry grants an Alter Self ability that allows usage of weapons, armour, all kind of thing, and even with 1 level in Shifter would be pretty easy to make work. Being an Alter Selfed man with swords and spellcasting abilities is generally superior to being an ooze, but sometimes people want ooze, so snoozing on the ooze is not what we'll.. dooze.
The querent also plays at a table where the rules are interpreted very literally to the point of what I would call silliness, and thus some of the below classes may not work due to precise wording or interpretations I do not have the anywhere near the patience to comb through.
Synthesist Summoner Solves A Lot Of Problems
It gives extra natural weapons (eclipsing the maximum 2 Morphic Weapons gives), grants a bunch of natural armour, and just generally brings you up to snuff combatwise with almost zero problems. You go quadruped, you get Pounce, and it's otherwise almost dealer's choice when picking abilities - as long as you're maxed out on natural attacks, you'll be a good but not overwhelming melee combatant with almost zero other influence.
If you're not interested in fusing with something with legs, it gets more complicated (and kinda awful)
The best way to get a reasonable AC while nekkid is to be a monk. Specifically, you probably want to be a Scaled or Nornkith monk and dump dex, pumping cha instead, because if you dip oracle (sidestep mystery) you can replace dex with cha to AC and effectively get your AC to cha twice. Then you cast Mage Armor (or get a party member to cast it). Buy a wand. Buy 3. It's a 1st level spell. Not like you have great ways to spend money anyway. You could also do that weird homunculus trick or w/e but getting literally any long duration spell infinitely for the cost of 10 potions would be shut down at most tables so i'm just gonna assume it's a wand.
Since you're investing in a decent charisma, you might as well be a Paladin and get that added to saves. Your attack routine at this point is unarmed strike + natural attacks which isn't too shoddy as you're largely str/cha/con, especially if you spend a feat on noble scion (war). It's still not great though.
Get a wand of Shield too while you're at it. Make people shove the wand in your oozy body and cast that on you all the time.
Regardless, you will probably want some vivisectionist alchemist.
Also known as the 'better rogue' (except for certain specific theft or fear builds), the vivisectionist alchemist is good for you for a few reasons. The first is that sneak attack applies to all your attacks and it's hard to get damage otherwise - you're not 2-handing anything or firing bows. The second is that extracts aren't spells, so you can pour those all over yourself repeatedly and that's fine and dandy according to Fluidic Body. There's not necessarily a lot at level 1 that helps you, but the 2nd level extracts are going to be very helpful like natural reach, bull's strength, etc.
The annoying part of this is that beastmorph or certain discoveries would grant natural attacks and they simply don't help because of how morphic weapons is written. You're best off simply not taking that archetype and just chugging your mutagen as is, or at most going cognatogen to further buff your cha. You should probably make sure to pick up Bottled Ooze though, because an ooze that bottles oozes and then throws the bottled oozes into combat is funny.
Oozemorph 1 ->
If more than dipping Summoner, Race -> Half-Elf.
Synthesist Summoner 1: Quadruped Form, Pounce, Natural Armour
Synthesist Summoner 4: 4 natural attacks allowed
Synthesist Summoner 5: 4 natural armour pre-buffs, str/dex bonus, 4 bab
Synth Summoner 9: 5th natural attack
Monk 1 (Scaled fist or Nornkith): Cha to AC while unarmoured
Oracle 1 (Sidestep Mystery): Cha to AC instead of Dex
Paladin 2: Cha to Saves
Vivisectionist Alchemist: No real limit or specific breakpoint here. Sneak attack, along with some means to enable it (blink, shatter defenses, what have you) is just nice and alchemist gets a bunch of really nice buffs whose casting mechanism bypasses the oozemorph 'no spells allowed' limitation. On top of that, it allows potion crafting with a great list of buffs which would allow further buffing and thus utility for the ooze. The fact that at level 6 you can craft oozes and hurl them at your foes is frankly the icing on the cake.
^ The above are all pieces to make an unarmed, no gear, no hands, melee character work. You can mix and match them as you like. While vivisectionist and summoner can be mixed, generally one should be prioritized as getting access to the spells both classes grant, specifically the longer-duration buffs, will be important for the character, and likely the main use of their oozemorph alter self ability.
Notably this kind of character is exactly the kind of character made far more interesting by two of the most used 3rd party pathfinder supplements - path of war and spheres of might. You can optimize a natural attack shredder or grappler lockdown or what have you in 1st party pf, but it's generally going to be a one trick wonder at most and have dead levels and generally be painful to get off the ground. Maneuvers and spheres both add a lot of options to a character like this without much muss or fuss and generally in a far more balanced way.